Observation of anomalous amplitude modes in the kagome metal CsV3Sb5

The kagome lattice provides a fertile platform to explore novel symmetry-breaking states. Charge-density wave (CDW) instabilities have been recently discovered in a new kagome metal family, commonly considered to arise from Fermi-surface instabilities. Here we report the observation of Raman-active CDW amplitude modes in CsV3Sb5, which are collective excitations typically thought to emerge out of frozen soft phonons, although phonon softening is elusive experimentally. The amplitude modes strongly hybridize with other superlattice modes, imparting them with clear temperature-dependent frequency shift and broadening, rarely seen in other known CDW materials. Both the mode mixing and the large amplitude mode frequencies suggest that the CDW exhibits the character of strong electron-phonon coupling, a regime in which phonon softening can cease to exist. Our work highlights the importance of the lattice degree of freedom in the CDW formation and points to the complex nature of the mechanism.

The symbols are analyzed values from Lorentzian peak fitting. The circles represent average values of the data points. Small variations are present due to experimental uncertainties. The standard deviation of the analyzed angle-dependent intensity divided by the average value yields the following results: 4.0% (E 2g , 4 K), 5.5% (E 2g , 100 K), 4.3% (A 1g , 4 K), and 5.3% (A 1g , 100 K). The slightly lower values at 4 K can be attributed to enhanced signal-to-noise ratio due to the strengthening of the main lattice phonon peaks, which do not support CDW-induced D 2h point group.  Figure 6. A detailed inspection of the A 2 mode. The Raman intensity color plot of (a) the raw data and (b) the data after subtracting the 130 K spectrum. (c) Spectra at 40-90 K (blue) and 120 K (orange) after subtracting the 130 K spectrum. The shaded blue highlights CDW-induced intensity, mainly from the A 2 mode that redshifts and broadens upon warming, but also from the A 1 mode at 45 cm −1 which does not shift with temperature.
Supplementary Figure 7. Imaginary phonon modes of pristine CsV 3 Sb 5 at three M points, M 1,2,3 . Because the phonon dynamical matrix eigenvector has a π phase at M , we plot it in a 2 × 2 supercell.

Supplementary Note 1: Chiral character of E 2g modes
For E 2g , the doubly degenerate modes correspond to states with opposite angular momentum (l = ±1), i.e. they are chiral. Although the dynamical matrix eigenvectors are usually exported as real vectors by suppressing l, we can reconstruct the chiral phonons by a projection operation to chiral states. For C 3 , the projection operator is defined as: is the eigenvalue of C 3 with pseudoangular momentum l = −1, 0, 1. For an arbitrary phonon dynamical matrix eigenvector u, the projected eigenvector P l u must be chiral, because

Supplementary Note 2: c-axis modulation
Below we consider Raman-active modes for different forms of c-axis modulation. At Γ point all phonon modes can be classified into acoustic and optic modes, i.e. Γ total = Γ acoustic ⊕ Γ optic . The optic phonon modes can be further classified into infra-red (IR) active, Raman active, and silent modes as Γ optic = Γ IR ⊕ Γ Raman ⊕ Γ silent . Based on group theory analysis [Kroumova et. al. Phase Transitions, 76, 155 (2003)], we have derived the symmetry representations of phonon modes for five different modulations along the c axis: These structures either have the D 6h or D 2h point groups. For the former (latter), the Raman-active modes that can be observed in the back-scattering geometry of our experiment are the A 1g and E 2g (A g and B 1g ) modes. Except for the 2 × 2 × 1 structure, the predicted number of modes far exceeds that observed experimentally. Therefore, in terms of Raman response, the c-axis modulation is not clearly manifested.  In principle, the D 6h and D 2h point groups can be distinguished by polarization angle dependent measurements. We consider the collinear polarization configuration (denoted as XX), in which the polarizations for the incident and scattered light are kept parallel while they are co-rotated with respect to a given crystal axis. For the D 6h point group, the A 1g and E 2g Raman-active modes that contribute to the back-scattering response have the following  Raman tensors, The Raman scattering intensity of both modes are expected to be independent of the angle of the linear polarization θ, I A 1g (θ) ∝ a 2 , I E 2g (θ) ∝ f 2 . For the D 2h point group, the relevant modes have A g and B 1g symmetries, and their Raman tensors are The Raman scattering intensity of both modes are expected to be anisotropic, I Ag (θ) ∝ (a cos 2 θ + b sin 2 θ) 2 , I B 1g (θ) ∝ d 2 [1 − cos(4θ)]/2. Supplementary Fig. 4 shows the polarization-angle dependent data of CsV 3 Sb 5 in the CDW phase (at 4 K) and the normal phase (at 100 K). We focus on the main lattice modes due to their excellent signal-to-noise ratio. Neither modes show appreciable polarization angle dependence, above and below the CDW transition. We therefore conclude that either the c-axis modulation is too weak to induce clear polarization angle dependence on the observed Raman modes, or those candidate stacking orders with the D 2h point group (ii and iv) can be ruled out, because the B 1g mode intensity should vary sharply between zero and the maximum value, which is incompatible with the observed results.
Supplementary Note 3: Decomposition of 3M + 1 modes We first calculated the character table of the three folded modes 3M + 1 (whose real space patterns are shown in Supplementary Figure 7) as shown in Supplementary Tab. 7. The representation of 3M + 1 is reducible and its decompositon onto each irreducible representation (irreps) can be done by standard textbook group theory method: where χ Γ i (g) refers to the character of the symmetry operator g belonging to irreps Γ i . Combining the characters of χ 3M + 1 (g) shown in Supplementary Tab. 7 and the irreps table of D 6h , we found that the only nonzero coefficients are: a A 1g = a E 2g = 1.